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Model predictive control of glucose concentration in type I diabetic patients:

An in silico trial

L. Magni a,*, D.M. Raimondo a, C. Dalla Man b, G. De Nicolao a, B. Kovatchev c, C. Cobelli b

a Dipartimento di Informatica e Sistemistica, University of Pavia, via Ferrata 1, I-27100 Pavia, Italyb Department of Information Engineering, University of Padova, Via Gradenigo 6/B, I-35131 Padova, Italyc University of Virginia Health System, Box 800137, Charlottesville, VA 22908, USA

1. Introduction

Diabetes (medically known as diabetes mellitus) is the name

given to disorders in which the body has trouble regulating its

blood glucose, or blood sugar, levels. There are two major types of

diabetes: type I diabetes and type II diabetes. Type I diabetes, also

called juvenile diabetes or insulin-dependent diabetes, occurs

when the body’s immune system attacks and destroys beta cells of

the pancreas. Beta cells normally produce insulin, a hormone that

regulates, together with glucagon (another hormone produced by

alpha cells of the pancreas), the blood glucose concentration in the

body, i.e. while insulin lowers it, glucagon increases it. The food

intake in the body results in an increase in the glucose

concentration. Insulin enables most of the body’s cells to take

up glucose from the blood and use it as energy. When the beta cells

aredestroyed, since no insulin can be produced,glucose remains in

the blood causing serious damage to several organs. For this

reason, people with type I diabetes must take insulin in order to

stay alive. This means undergoing multiple injections daily

(normally coinciding with the meal times), and testing blood

sugar by pricking their fingers for blood several times a day [8].

This method is just an open-loop control with intermittently

monitored glucose levels resulting in intermittently administered

insulin doses that are manually administered by a patient using awritten algorithm. The current open-loop therapy can be

contrasted to glucose management by a closed-loop system

controlling blood glucose in diabetic patients also known as an

artificial pancreas.

The automated control of normoglycemia in subjects with type

I diabetes has been subject of extensive research since the 1970s

[1,7]. However, the first devices, e.g. the Biostator TM [4], which

used intravenous (i.v.) BG sampling and i.v. insulin and glucose

delivery, were cumbersome and unsuited for outpatient use. A

minimally invasive closed-loop system using subcutaneous (s.c.)

continuous glucose monitoring and s.c. insulin pump delivery was

needed. To date, several s:c :– s:c : systems, have been tested [7,9].

From a control viewpoint, the main challenges are time delays,

constraints, meal disturbances, and nonlinear dynamics. Most of

the control schemes proposed in literature are based on either PID

(proportional integral derivative) or MPC (model predictive

control) control laws, see e.g. [16,7,6]. Due to the presence of

significant delays in the glucose metabolism model, a well-

designed feedforward action able to consider in advance a meal

announcement signal is essential in order to guarantee satisfactory

performances. Within MPC , feedforward and feedback control

actions are jointly designed and constraints are taken into account

in a very natural way. The recent advancements in nonlinear MPC ,

see e.g. [15], and the development of a new generationof models of

glucose metabolism make this control technique even more

promising.

Biomedical Signal Processing and Control 4 (2009) 338–346

A R T I C L E I N F O

Article history:Received 10 November 2008

Received in revised form 6 April 2009

Accepted 15 April 2009

Available online 22 May 2009

Keywords:

Diabetes

Virtual patients generation

Model predictive control

Control-Variability Grid Analysis

Blood Glucose Index

A B S T R A C T

In this paper, the feedback control of glucose concentration in type I diabetic patients usingsubcutaneous insulin delivery and subcutaneous continuous glucose monitoring is considered. A

recently developed in silico model of glucose metabolism is employed to generate virtual patients on

which control algorithms can be validated against interindividual variability. An in silico trial consisting

of 100 patients is used to assess the performances of a linear output feedback and a nonlinear state-

feedback model predictive controller, designed on thebasis of thein silico model.More than satisfactory

results areobtainedin thegreatmajority of virtual patients.The experiments highlight thecrucialrole of

the anticipative feedforward action driven by the meal announcement information. Preliminary results

indicate that further improvements may be achieved by means of a nonlinear model predictive control

scheme.

2009 Elsevier Ltd. All rights reserved.

* Corresponding author.

E-mail address: [email protected] (L. Magni).

Contents lists available at ScienceDirect

Biomedical Signal Processing and Control

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / b s p c

1746-8094/$ – see front matter 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.bspc.2009.04.003

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The need of guaranteeing parameter identificability from easily

measurable experimental data motivated the development of

parsimonious models of glucose metabolism (see e.g. the classical

minimal model [2]). By contrast, in silico large-scale simulation

models aim to describe the glucose–insulin system in as much

detail as possible. The major limitation of most in silico models is

that they were validated on few subjects and only by means of

plasma concentration measurements. Recently, a new-generation

in silico model has been developed taking advantage of the

availability of a unique-meal data set of 204 nondiabetic

individuals [5]. The subjects underwent a triple tracer meal

protocol, making it possible to obtain in a virtually model-

independent fashion thetime courseof all therelevant glucose and

insulin fluxes during a meal. Thus, by using a ‘‘concentration and

flux’’ portrait, it was possible to model the glucose–insulin system

by resorting to a subsystem forcing function strategy which

minimizes structural uncertainties in modeling the various unit

processes.

In this paper, the above model is used to randomly generate

virtual diabetic patients. Suitable modifications (see [12]) are

introduced to reflect the absence of endogenous insulin secretion,

while the parameter variability in the original data set is used to

model interindividual variability of the diabetic population.

Validating algorithms on a whole set of different patients (insilico trial) is the only realistic way of addressing robustness of the

artificial pancreas in the face of interindividual variability so as to

maximize success chances in the subsequent clinical trials

conducted on real patients.

In the present work, the in silico model is used todesign a linear

output feedback MPC scheme which is then tested on an in silico

trial consisting of 100 synthetic type I diabetic subjects ‘‘followed’’

for a day, receiving dinner and breakfast. The overall results, as

measured by several established performance indices, are more

than satisfactory for the great majority of virtual patients. An

important role is played by the feedforward action that takes

advantage of the meal announcement information. Moreover, the

controller performances are shown to be robust with respect to

errors in the meal announcement signal. Finally, in order to furtherimprove the glucose concentration regulation, a preliminary test is

carried out by using a nonlinear state-feedback MPC scheme. A

definite improvement of glucose and insulin profiles is observed,

indicating that nonlinear MPC has the potential for achieving very

effective glycemic control.

2. Virtual patients

In order to synthesize and test the controller, we used the meal

glucose–insulin model [5,12] summarized in the following

subsections.

2.1. Glucose intestinal absorption

Glucose intestinal absorption is modeled by a three-compart-

ment model

˙ Q sto1ðt Þ ¼ k griQ sto1ðt Þ þ dðt Þ

˙ Q sto2ðt Þ ¼ kempt ðt ;Q stoðt ÞÞQ sto2ðt Þ þ k griQ sto1ðt Þ

˙ Q gut ðt Þ ¼ kabsQ gut ðt Þ þ kempt ðt ;Q stoðt ÞÞQ sto2ðt Þ

Q stoðt Þ ¼ Q sto1ðt Þ þ Q sto2ðt Þ

Raðt Þ ¼ fkabsQ gut ðt Þ=BW

8>>>>>><>>>>>>:where Q sto (mg) is the amount of glucose in the stomach (solid,

Q sto1, and liquid phase, Q sto2

), Q gut (mg) is the glucose mass in the

intestine, k gri is the rate of grinding, kabs is the rate constant of

intestinal absorption, f is the fraction of intestinal absorption

which actually appears in plasma, d (mg/min) is the amount of

ingested glucose, BW (kg)is thebody weight, Ra (mg/kg/min) is the

glucose rate of appearance in plasma and kempt is the rate constant

of gastric emptying which is a time-varying nonlinear function of

Q sto

kempt ðt ; Q stoðt ÞÞ ¼ kmax þkmax kmin

2 ftan h½aðQ stoðt Þ bDðt ÞÞ

tan h½bðQ stoðt Þ dDðt ÞÞg

where

a ¼ 5

2Dðt Þð1 bÞ; b ¼

5

2Dðt Þd; Dðt Þ ¼

Z t f

t i

dðt Þ dt

with t i and t f , respectively, start time andend time of thelast meal,

b, d , kmax and kmin model parameters.

2.2. Glucose subsystem

A two-compartment model is used to describe glucose kinetics

˙ G pðt Þ ¼ k1G pðt Þ þ k2Gt ðt Þ þ EGP ðt Þ þ Raðt Þ U ii E ðt Þ

˙ Gt ðt Þ ¼ k1G pðt Þ k2Gt U idðt Þ

Gðt Þ ¼ G pðt Þ

V G

8>>><

>>>:where V G (dl/kg) is the distribution volume of glucose, G (mg/dl) is

the glycemia, G p (mg/kg) and Gt (mg/kg) are glucose in plasma and

rapidly equilibrating tissues, and in slowly equilibrating tissues,

respectively, EGP is the endogenous glucose production (mg/kg/

min), E (mg/kg/min) is the renal excretion, U ii and U id are the

insulin-independent and -dependent glucose utilizations, respec-

tively (mg/kg/min), and k1 and k2 are the rate parameters. The

insulin-independent glucose utilizations U ii is assumed constant.

Basal steady state, i.e. constant glycemia Gb (mg/dl), is character-

ized by the following equations:

k1G pb þ k2Gtb þ EGP b þ Rab U ii E b ¼ 0k1G pb k2Gtb U idb ¼ 0

so that, noting that at basal equilibrium Rab ¼ 0,

EGP b ¼ U ii þ U idb þ E b

2.3. Glucose renal excretion

Renal extraction represents the glucose flow which is elimi-

nated by thekidney, when glycemia exceeds a certain threshold ke2

E ðt Þ ¼ max f0; ke1ðG pðt Þ ke2Þg

The parameter ke1 (1/min) represents renal glomerular filtration

rate.

2.4. Endogenous glucose production

EGP comes from the liver, where a glucose reserve exists

(glycogen). EGP is inhibited by high levels of glucose and insulin

EGP ðt Þ ¼ max ð0; EGP b k p2ðG pðt Þ G pbÞ k p3ðI dðt Þ I bÞÞ

where k p2 and k p3 are model parameters and I d (pmol/l) is a

delayed insulin signal, coming from the following dynamic

system:

˙ I 1ðt Þ ¼ kiI ðt Þ kiI 1ðt Þ˙ I dðt Þ ¼ kiI 1ðt Þ kiI dðt Þ

( (1)

where I (pmol/l) is plasma insulin concentration or insulinemia

and k i (1/min) is a model parameter.

L. Magni et al./ Biomedical Signal Processing and Control 4 (2009) 338–346 339



2.5. Glucose utilization

Glucose utilization is made up of two components: the insulin-

independent one U ii, which represents the glucose uptake by the

brain and erythrocytes, and the insulin-dependent component U id,

which depends nonlinearly on glucose in the tissues

U idðt Þ ¼ V mð X ðt ÞÞ Gt ðt Þ

K m þ Gt ðt Þ

where V m (1/min) is a linear function of interstitial fluid insulin X (pmol/l)

V mð X ðt ÞÞ ¼ V m0 þ V mx X ðt Þ

which depends from insulinemia in the following way:

˙ X ðt Þ ¼ p2U X ðt Þ þ p2U ðI ðt Þ I bÞ

where K m, V m0, V mx are model parameters, I b (pmol/l) is the basal

insulin level and p2U (1/min) is called rate of insulin action on

peripheral glucose. Considering the basal steady state, one has

U idb ¼ V m0Gtb

K m þ Gtb

and

Gtb ¼ U ii EGP b þ k1G pb þ E bk2

V m0 ¼ ðEGP b U ii E bÞðK m þ GtbÞ

Gtb

2.6. Insulin subsystem

Insulin flow s, coming from the subcutaneous compartments,

enters the bloodstream and is degradated in the liver and in the

periphery:

˙ I pðt Þ ¼ ðm2 þ m4ÞI pðt Þ þ m1I lðt Þ þ sðt Þ

˙ I lðt Þ ¼ ðm1 þ m3ÞI lðt Þ þ m2I pðt Þ

I ðt Þ ¼ I pðt Þ=V I

8><

>:where V I (l/kg) is thedistribution volumeof insulin and m1, m2, m3,m4 (1/min) are model parameters.

At basal one has

0 ¼ ðm2 þ m4ÞI pb þ m1I lb þ sb

0 ¼ ðm1 þ m3ÞI lb þ m2I pb

so that

I lb ¼ m2

m1 þ m3I pb

sb ¼ ðm2 þ m4ÞI pb m1I lb

8<:where I pb ¼ I bV I .

2.7. s.c. insulin subsystem

In diabetic patients, insulin is usually administered by

subcutaneous injection. Insulin takes some time to reach the

circulatory apparatus, unlike in the healthy subject in which the

pancreas secretes directly into the portal vein. This delay is

modeled here with two compartments, S 1 and S 2 (pmol/kg) (which

is a variation of the model described in [17]), which represent,

respectively, polymeric and monomeric insulin in the subcuta-

neous tissue

˙ S 1ðt Þ ¼ ðka1 þ kdÞS 1ðt Þ þ uðt Þ˙ S 2ðt Þ ¼ kdS 1ðt Þ ka2S 2ðt Þ

sðt Þ ¼ ka1S 1ðt Þ þ ka2S 2ðt Þ

8><>:where uðt Þ (pmol/kg/min) represents injected insulin flow, kd is

called degradation constant, ka1 and ka2 are absorption constants.

At basal one has

0 ¼ ðka1 þ kdÞS 1b þ ub

0 ¼ kdS 1b ka2S 2b

sb ¼ ka1S 1b þ ka2S 2b

8><>:and solving the system

S 1b ¼ sb

kd þ ka1

S 2b ¼ kd

ka1S 1b

ub ¼ sb

8>>>><>>>>:The quantity ub (pmol/min) represents insulin infusion to maintain

diabetic patient at basal steady state.

2.8. s.c. glucose subsystem

Subcutaneous glucose GM (mg/dl) is, at steady state, highly

correlated with plasma glucose; dynamically, instead, it follows

the changes in plasma glucose with some delay. This dynamic was

modeled with a system of first order

˙ GM ðt Þ ¼ ks:c :GM ðt Þ þ ks:c :Gðt Þ

2.9. Virtual patient generation

In order to obtain parameter joint distributions in type 1

diabetes, the parameter identified in 204 subjects in health were

used as starting point [5]. Some modification was needed to

realistically describe a type 1 diabetic subject: basal glucose

concentration was assumed to be on average 50 mg/dl higher

than in subjects in health; steady-state insulin concentration

(due to an external insulin pump) was assumed to be on average

four times higher than in subjects in health; basal endogenous

glucose production was assumed to be 35% higher than in

subjects in health, and steady-state insulin clearance was

assumed to be one third lower than in subjects in health;

parameters relating to insulin action on both glucose productionand utilization were assumed to be one-third lower than in

subjects in health. For all these parameters/variables the same

inter-subject variability found in subjects in health was

maintained. The parameters were assumed to be log-normal

distributed to guarantee that they were always positive, so the

covariance matrix (26 26) was calculated using the log-

transformed parameters. Then, 100 subjects were generated

using the joint distribution, i.e. 100 realizations of the log-

transformed parameter vector were randomly extracted from

the multivariate normal distribution with mean equal to mean

of the log-transformed parameters and 26 26 covariance

matrix. Finally, the parameters in the 100 in silico subjects

were obtained by antitransformation.

3. Metrics to assess control performance

In evaluating the performance of a control algorithm one has

to remember that its basic function is to mimic as best as

possible the feature of the beta cells, which is to maintain

glucose levels between 80 and 120 mg/dl in the face of

disturbances such as meals or physical activity. A good

algorithm should be able to maintain blood sugar low enough,

as this reduces the long-term complications related to diabetes,

but must also avoid even isolated hypoglycemic episodes. To be

useful, a metric has to take into account these features. The

HbA1c cannot be considered in this context because in the first

closed-loop clinical trial patients are controlled only for a short

period.




3.1. Blood Glucose Index (BGI)

Blood Glucose Index is a metric proposed by Kovatchev et al.

[10], to evaluate the clinical risk related to a particular glycemic

value

BGI ðÞ ¼ 10ð g ½lna

ðÞ bÞ2

(2)

where a, b and g are fixed equal to 1.084, 5.3811 and 1.509,

respectively. The BGI function is asymmetric, because hypoglyce-mia is considered more dangerous than hyperglycemia (see Fig. 1).

Starting from the BGI, two synthetic indices for sequences of n

measurements of blood glucose can be defined.

3.1.1. LBGI (low BGI)

Measures hypoglycemic risk

LBGI ¼ 10

n

Xn

i¼1

rl2

ðGiÞ

where rlðÞ ¼ min ð0; g ½lna

ðÞ lna

ðG0ÞÞ.

3.1.2. HBGI (high BGI)

Measures hyperglycemic risk

HBGI ¼ 10

n

Xn

i¼1

rh2

ðGiÞ

where rhðÞ ¼ max ð0; g ½ln aðÞ ln aðG0ÞÞ.

LBGI index captures the tendency of the algorithm to overshoot

the target and eventually trigger hypoglycemia. Directly linked

with LBGI, HBGI captures the tendency of the algorithm to stay

above the target range.

3.2. Control-Variability Grid Analysis (CVGA)

The Control-VariabilityGrid Analysis (CVGA) [13] is a graphical

representation of min/max glucose values in a population of

patients either real or virtual. The CVGA provides a simultaneous

assessment of thequalityof glycemic regulation in all patients. Assuch, it hasthe potential to play an important role in thetuning of

closed-loopglucosecontrol algorithms andalso in thecomparison

of their performances. Assumingthat foreach subject a timeseries

of measured blood glucose (BG) values over a specified time

period (e.g.1 day) is available, theCVGAis obtainedas follows:for

each subject, a point is plotted whose X -coordinate is the

minimum BG andwhose Y -coordinate is the maximum BG within

the considered time period (see Fig. 2). Note that the X -axis is

reversed as it goes from 110 (left) to 50 (right) so that optimal

regulation is located in lower left corner. The appearance of the

overall plot is a cloud of points located in various regions of the

plane. Different regions on the plane can be associated with

different qualities of glycemic regulation. In order to classify

subjects into categories, nine rectangular zones are defined as

follows:

Zone A(Gmax < 180, Gmin > 90)]: Accurate control.

Zone B high (180 <Gmax < 300, Gmin > 90)]: Benign trend

towards hyperglycemia.

Zone B low (Gmax <180, 70<Gmin < 90)]: Benign trend towards

hypoglycemia.

Zone B (180<Gmax < 300, 70 <Gmin < 90)]: Benign control.Zone C high (Gmax > 300, Gmin > 90)]: overCorrection of

hypoglycemia.

Zone C low (Gmax < 180, Gmin < 70)]: overCorrection of hyper-

glycemia

Zone D high (Gmax > 300, 70<Gmin < 90)]: Failure to deal with

hyperglycemia.

Zone D low (180 <Gmax < 300, Gmin < 70)]: Failure to deal with

hypoglycemia.

Zone E (Gmax >300, Gmin < 70)]: Erroneous control.

4. Model predictive control

The glucose metabolism model can be rewritten in the

following compact way:˙ xðt Þ ¼ f ðt ; xðt Þ;uðt Þ;dðt ÞÞ

yðt Þ ¼ GM ðt Þ

where x ¼ ½Q sto1; Q sto2

;Q gut ;G p;Gt ; I p; X ; I 1; I d; I l; S 1; S 2; GM , and

f ð; ; ; Þ is derived from the model equations reported in Section

2. The system is subject to the following constraints:

xmin x xmax (3)

umin u umax (4)

where xmin, xmax, umin and umax denote lower and upper bounds on

the state and input, respectively. Typically, they represent limits on

the glucose concentration and on the insulin delivery rate. In theFig. 1. Blood Glucose Index.

Fig. 2. Control-Variability Grid Analysis with summary.




following, it is assumed that meal announcement is available, i.e.

the disturbance signal (the meal) is known in advance.

The MPC control law ([15,3,11]) is based on the solution of a

Finite Horizon Optimal Control Problem (FHOCP ), where a cost

function J ð ¯ x;uÞ is minimized with respect to the input u subject to

the state dynamics of a model of the system, and to state- and

input-constraints. Letting uo be the solution of the FHOCP ,

according to the Receding Horizon paradigm, the feedback control

law u ¼ kMPC ð xÞ is obtained by applying to the system only the first

part of the optimal solution. In this way, a closed-loop control

strategy is obtained solving an open-loop optimization problem.

MPC control laws can be formulated for both discrete- and

continuous-time systems. In this paper, a discrete-time linear MPC

(LMPC ) is derived from an input–output linearized approximation

of the full model. Moreover, a state-feedback nonlinear MPC

(NMPC ) scheme is derived that exploits the full nonlinear model.

4.1. Unconstrained linear model predictive control

In order to obtain a time-invariant model, the rate constant

of gastric emptying kempt has been fixed to kempt ðt ;Q stoÞ ¼ kmean ¼

ðkmin þ kmaxÞ=2. The system is linearized around the basal steady-

state point of each patient and discretized with sampling time T s

yielding

xðk þ 1Þ ¼ AD xðkÞ þ BDuuðkÞ þ BDddðkÞ yðkÞ ¼ C D xðkÞ

After a model order reduction step the following transfer function

representation is obtained

Y ð z Þ ¼ N U ð z Þ

DE ð z ÞU ð z Þ þ

N Dð z Þ

DE ð z ÞDð z Þ (5)

with

N U ð z Þ ¼ b2 z 2 þ b1 z þ b0

DE ð z Þ ¼ z 3 þ a2 z 2 þ a1 þ a0

N Dð z Þ ¼ bD2 z 2 þ bD1 z þ bD0

Since the two transfer functions have the same denominator, the

following input–output representation is obtained

yðk þ 1Þ ¼ a2 yðkÞ a1 yðk 1Þ a0 yðk 2Þ þ b2uðkÞ þ b1uðk 1Þ

þ b0uðk 2Þ þ bD2dðkÞ þ bD1dðk 1Þ þ bD0dðk 2Þ

Finally, system (5) can be given the following state-space

(nonminimal) representation

xIOðk þ 1Þ ¼ AIO xIOðkÞ þ BIOuðkÞ þ M IOdðkÞ yðkÞ ¼ C IO xIOðkÞ

where xIOðk þ 1Þ ¼ ½ yðk þ 1Þ0; . . . ; yðk n þ 2Þ

0;uðkÞ

0; . . . ;uðk n

þ2Þ0;dðkÞ0; . . . ; dðk n þ 2Þ

00 and the matrices A IO;BIO;M IO;C IO are

defined accordingly.

In order to derive the LMPC control law the following quadraticdiscrete-time cost function is considered

J ð xIOðkÞ; uðÞÞ ¼XN 1

i¼0

ðk yoðk þ iÞ yðk þ iÞk2Q D

þ kuðk þ iÞk2RD

Þ

þ k yoðk þ N Þ yðk þ N Þk2S D

where N is the prediction horizon, yoðkÞ the desired output at time

k and Q D ¼ Q 0D 0, RD ¼ R0D > 0, S D ¼ S 0D 0 areweighting matrices.

The evolution of the system can be rewritten in a compact way

as follows:

Y ðkÞ ¼ A c xIOðkÞ þ B c U ðkÞ þ Mc DðkÞ

where Y ðkÞ ¼ ½ yðk þ 1Þ; yðk þ 2Þ; . . . ; yðk þ N 1Þ; yðk þ N Þ;DðkÞ ¼

½dðkÞ

0

; dðk þ 1Þ

0

;. . .

;dðk þ N 1Þ

0

; dðk þ N Þ

0

0

; U ðkÞ ¼ ½uðkÞ

0

; uðk þ 1Þ

0

;

. . . ; uðk þ N 1Þ0;uðk þ N Þ

00 and A c ; B c ;Mc , are derived using the

discrete-time Lagrange formula:

xIOðk þ iÞ ¼ AiIO xIOðkÞ þ

Xi1

j¼0

Ai j1IO ðBIOuðk þ jÞ þ M IOdðk þ jÞÞ

In this way, letting Y oðkÞ ¼ ½ yoðk þ 1Þ0; yoðk þ 2Þ

0; . . . ; yoðk þ N 1Þ

0;

yoðk þ N Þ00 the cost function becomes

¯ J ð xðkÞ; uðÞÞ ¼ ðY oðkÞ Y ðkÞÞ0QðY oðkÞ Y ðkÞÞ þ U 0ðkÞRU ðkÞ

¼ ðY oðkÞ A c xIOðkÞ B c U ðkÞ Mc DðkÞÞ0Q ðY oðkÞ

A c xIOðkÞ B c U ðkÞ Mc DðkÞÞ þ U 0ðkÞRU ðkÞ

where Q and R are block-diagonal matrices that contain the

matrices Q , R and S . In the unconstrained case, the solution of the

optimization problem is

U oðkÞ ¼ ðB 0c QB 0c þ RÞ1

B 0c QðY oðkÞ A c xIOðkÞ Mc DðkÞÞ

where the future values of the set point and the disturbance signal

(meal) are considered. Finally, following the Receding Horizon

approach the control law is given by

uoðkÞ ¼ ½ I 0 . . . 0 U oðkÞ

In view of the input constraints (4), only a saturated value will be

applied to the system. The satisfaction of the state constraints, on

the contrary, cannot be guaranteed; it is only possible to tune

parameters Q , R , S and N to improve the regulation performance.

The major advantages of this input–output LMPC scheme are that

an observer is not required ( xIO includes past input and output

values only), and that it is very easily implementable because on-

line optimization is avoided.

4.2. Constrained linear model predictive control

Witha relatively small increaseof the computational burden it is

possible to consider explicitly both input and state constraints by

solving a constrained linear quadratic optimization problem. This

can be doneby solving on-line a quadratic programmingproblem or

by using an explicit solution derived through a multiparametric

approach. In this paper the results obtained with constrained LMPC are not reported because they did not show any significant

improvement. In fact, the explicit consideration of input constraints

alone does not improve the performance of the unconstrained

saturated control law, while the fully constrained problem, i.e. also

with state constraints, introduces nontrivial problems due to the

approximation error caused by linearization and model reduction.

Further work is required to explore this approach.

4.3. Nonlinear model predictive control

Unconstrained LMPC with a quadratic cost function provides an

explicit control law. The simplicity of this solution is a great

advantage in order to implement the controller on real patients.

In the following, a nonlinear model predictive control isproposed. The goal is to evaluate the (upper) performance limit

of a predictive control algorithm in the blood glucose control

problem. In order to do this, we consider the ideal albeit

unrealistic, situation:

1. the nonlinear model used in the synthesis of the NMPC exactly

describes the system;

2. all the states of the systems are measurable. A state-feedback

NMPC is proposed (no observer is used).

The model used in the synthesis of the NMPC is the full

nonlinear continuous-time model previously described.

Since nonlinear predictive control allows for the use of

nonlinear cost functions, the adopted cost function incorporates




the kind of nonlinearity entering the LBGI and HBGI indexes

previously described. In order to minimize the control energy

spent within the optimization horizon, the cost function includes

also a term depending on the control action.

The Finite Horizon Optimal Control Problem consists in

minimizing, with respect to control sequence u1ðt kÞ; . . . ;uN ðt kÞ

(t k ¼ kT S with T S sampling time period), the following nonlinear

cost function with horizon N

J ð xt k ;u1ðt kÞ; . . . ; uN ðt kÞÞ ¼

Z t kþN

t k

f10qð g ððln ðG pðt ÞÞÞa bÞÞ2

þ ruðt Þ2g dt (6)

Note that a full hybrid solution is developed, i.e. the control inputs

uiðt kÞ are constrained to be piecewise constants, uðt Þ ¼

uiðt kÞ;t 2 ½t kþi1; t kþiÞ; i 2 ½1; . . . ; N (see [14]).

The values of g , a and b adopted in this case are g ¼ 0:75,

a ¼ 1:44, and b ¼ 9:02.

5. Virtual protocol

The performance of closed-loop glucose controllers was testedon a 1-day virtual protocol (see Fig. 3):

1. Patient enters at 17.00 of Day 1. At that time, data gathering for

warm-up phase starts. Microinfusor is programmed to inject

basal insulin for the specific patient.

2. At 18.00 of Day 1 patient takes a meal lasting about 15 min. The

meal contains 85 g of carbohydrates. At the same time, patient

receives an insulin bolus d according to his personal insulin/

carbohydrate ratio (CHO ratio or CR), that is

d ¼ 85 CR

3. Control loop is closed at 21.30 of Day 1. At this time, the

switching phase starts. In this phase the metrics for evaluationare not calculated.

4. At 23.00 of Day 1 regulation phase starts.

5. At 7.30 of Day 2, patient has breakfast containing 50 g of CHO.

Meal duration is about 2 min.

6. Experiment finishes at 12.00 of Day 2. Patient is discharged and

restarts his normal insulinic therapy.

6. In silico trial

In order to test the proposed control algorithm several in

silico trials have been based on the protocol described in Section

5. The main objectives are to verify: (a) the effect of a change in

the parameter q that is the only tuning parameter that is

individualized; (b) the impact of meal announcement; (c) therobustness with respect to errors in the meal announcement

information (e.g. time and amount); (d) the benefit of an explicit

consideration of the nonlinear dynamics by means of a NMPC.

The CVGA matrix, used to compare population performances, is

computed only during the regulation period in order to reduce

Fig. 5. CVGA: Experiment L1 (white) versus Experiment L3 (blue). (For

interpretation of the references to color in this figure legend, the reader is

referred to the web version of the article.)

Fig. 3. Nominal scenario for adult patient.







the influence of the open-loop regulation on the closed-loop

indexes:

Experiment L1. 100 subjects are simulated using an LMPC

control law synthesized with T s ¼ 15 min, n ¼ 3, N ¼ 16, RD ¼ 1

and Q D ¼ S D ¼ qn, where qn has been individualized for each

subject and the set point is taken equal to 112 mg/dl.

Experiment L2. It is equal to Experiment 1 with q ¼ 2qn.

Experiment L3. It is equal to Experiment 1, but the LMPC

control law is applied without meal announcement.

Experiment L4. The ingested amount of glucose is 125% of the

nominal value forall 100 patients.The LMPC control lawhas the

same parameters as those used for Experiment 1 and relies on

the nominal glucose dose to decide the feedforward action.

Experiment L5. The ingested amount of glucose is 75% of thenominal value forall 100 patients.The LMPC control lawhas the

same parameters as those used for Experiment 1 and relies on

the nominal glucose dose to decide the feedforward action.

Experiment L6. The ingested amount of glucose is delayed of

30 min with respect to the nominal value for all 100 patients.

The LMPC control law has the same parameters as those used

for Experiment 1 and relies on the nominal glucose profile to

decide the feedforward action.

Experiment L7. The ingested amount of glucose is 30 min in

advance with respect to the nominal value for all 100 patients.

The LMPC control law has the same parameters as those used

for Experiment 1 and relies on the nominal glucose profile to

decide the feedforward action.

Experiment N8. In order to explore the potentiality of NMPC,

some virtual patients undergoing Experiment 1 were also

controlled by means of NMPC.

7. Results

The performances obtained in Experiment L1, as shown in Fig. 4,are clearly satisfactory: in factno hypo phenomenon occurs and also

themaximum values arevery well regulated.In particular, 59 virtual

patients are in the A-zone and41 are in the B-zone. The comparison

between Experiment L1 and L2 (see Fig. 4) shows the effect of a

variation of the q parameter, the only parameter to be tuned. In
















Fig. 10. Experiments L1 and L4–L7 for patient 9.

Fig. 11. Experiment L1 (LMPC) versus Experiment N8 (NMPC) for patient 35.




particular, increasing the aggressiveness, i.e. using a larger q, all the

points in the CVGA move towards the Lower B region reducing the

maximum BG value butalso theminimum one.2 patients move even

in the lower D-zone and 2 in thelower C-zone. The CVGA reported in

Fig. 5 for ExperimentsL1, L3 shows that in Experiment L3, 9 patients

move from A-zone to B-zone. The difference in regulation

performance with and without meal announcement information

demonstrates the benefit of an anticipative action in the insulin

injection exploiting knowledge of the meal profile. The robustness

analysis of the proposed control algorithm against wrong informa-

tion on meal amount or time is performed through Experiments L4–

L7andis shown in Figs. 6–9 . Inall cases, all the patients are in the A-

and B-zones so that. Robustness with respect to this kind of

uncertainties is apparent. An example of the time evolution of the

glucoseand external insulinare reported in Fig.10 forpatient #9.The

vertical black lines in Figs. 10 and 11 indicate the beginning of the

closed-loop, the beginning of the regulation period and the breakfast

nominal time. Finally, in Fig. 11 the glucose and insulin profiles

obtained in the Experiment L1 with the LMPC andin Experiment N8

with NMPC are reported for patient #35. As shown in Fig. 11, the

adoption of an NMPC scheme, in place of an LMPC one, may bring

substantial improvement. In particular the range of glucose

concentration variability is reduced and a much smoother insulin

profileis achieved. Theadvantage of the nonlinear predictivecontrolscheme is twofold: constraints on glycemia areexplicitly allowedfor

and knowledge of the nonlinear dynamics is exploited.

8. Conclusions

This paper presents an engineering view of the control of

diabetes based on three fundamental clinical perspectives: (1)

recent studies have documented thebenefits andthe technological

reliability of continuous glucose monitoring (CGM ); (2) intensive

insulin treatment attempting to approximate near-normal levels

of glycemia markedly reduces chronic complications, but may risk

potentially life-threatening severe hypoglycemia; (3) in silico

modeling could produce credible pre-clinical information and

could substitute the traditional animal trials yielding results in afraction of the time. Hence, the future development of the artificial

pancreas will be greatly accelerated by employing mathematical

modeling and computer simulation.

The best way to utilize CGM data would be to devise automated

procedures for the interpretation and processing of the continuous

data flow,which take intoaccountglucoseutilizationand the delays

related to subcutaneous insulin administration. Automated control

algorithms would be needed,which should be based on engineering

understanding of the glucose–insulin system in diabetes, e.g. on

predictive models. Risk analysis and analysis of temporal glucose

variability, including methods such as the BGI and the CVGA

presented in this paper, will be fundamental to both the design of

open- and closed-loop control algorithms and to the evaluation of

the results from their in silico and in vivo testing. A simulator of T 1DM , as the oneconsidered in this paper, shouldbe equipped with a

cohort of i n silico subjects that span the observed interindividual

variability of key metabolic parameters in the T 1DM population.

The in silico trial has demonstrated that linear output-feedback

MPC achieves satisfactory glycemic regulation in a virtual

population of 100 type I diabetic patients. The proposed scheme

is also robust with respect to errors in the meal announcement

signal. Robustness in the face of sensor errors may also be

investigated by complementing the simulator with a probabilistic

model of sensor noise. Moreover, the benefit brought by

individualization of the control parameter q has been demon-

strated. For virtual patients, this parameter can be tuned via a trial

and error procedure based on in silico experiments. For the real

patients it is necessary to find a safe value that can be refined via a

day-to-day iterative learning algorithm.

Preliminary tests conducted on virtual patients have shown

that state-feedback nonlinear MPC has the potential to introduce

further improvements. In fact, although linear MPC yields well-

regulated glucose profiles with reasonable insulin administrations,

the adoption of the nonlinear controller further reduces glucose

variability using a smoother insulin profile. However, for nonlinear

MPC to be usable in practice the development of state observers

would be required.

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